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4.2. The luminosity distance and gravitational lensing

Before proceeding to discuss possible constraints on OmegaLambda from gravitational lensing in this section and high redshift supernovae in the next, let us introduce a quantity which plays a crucial role in these discussions, namely the luminosity distance dL(z) up to a given redshift z. Consider an object of absolute luminosity curlyL located at a coordinate distance r from an observer at r = 0. Light emitted by the object at a time t is received by the observer at t = t0, t and t0 being related by the cosmological redshift 1 + z = a(t0) / a(t). The luminosity flux reaching the observer is

Equation 22 (22)

where dL is the luminosity distance to the object [142]

Equation 23 (23)

The luminosity distance dL depends sensitively upon both the spatial curvature and the expansion dynamics of the universe. To demonstrate this we determine dL using the expression for the coordinate distance r obtained by setting ds2 = 0 in (2), resulting in

Equation 24 (24)

which gives

Equation 25 (25)

where eta = integ0t cdt / a(t).

Furthermore, since dz / dt = - (1 + z)H(z), we get

Equation 26 (26)

where h(z) = H(z) / H0 is defined in (18), and, in a universe with several components

Equation 27 (27)

Substituting (27) and (26) in (23) we get the following expression for the luminosity distance in a multicomponent universe with a cosmological term [26]

Equation 28 (28)


Equation 29 (29)

and S(x) is defined as follows: S(x) = sin(x) if kappa = 1 (Omegatotal > 1), S(x) = sinh(x) if kappa = - 1 (Omegatotal < 1), S(x) = x if kappa = 0 (Omegatotal = 1).

Figure 4

Figure 4. The luminosity distance dL (in units of H0-1) is shown as a function of cosmological redshift z for flat cosmological models with a cosmological constant Omegam + OmegaLambda = 1. Heavier lines correspond to larger values of Omegam. For comparison we also show (dashed line) the angular size in a flat de Sitter universe (OmegaLambda = 1).

Before we turn to applications, let us consider a simple example which provides us with an insight into the role played by the luminosity distance dL in cosmology. In a spatially flat universe the expression for dL simplifies considerably, so that we get for the matter dominated model (a propto t2/3):

Equation 30 (30)

On the other hand in de Sitter space (a propto exp(H0 t))

Equation 31 (31)

Comparing (30) and (31) we find dLDS(z) > dLMD(z), which means that an object located at a fixed redshift will appear brighter in an Einstein-de Sitter universe than it will in de Sitter space (equivalently in the steady state model). (5) This is also true for a two component universe consisting of matter and a cosmological constant as demonstrated in Fig 4. In a spatially flat universe the presence of a Lambda-term increases the luminosity distance to a given redshift, leading to interesting astrophysical consequences. Since the physical volume associated with a unit redshift interval increases in models with Lambda > 0, the likelihood that light from a quasar will encounter a lensing galaxy is larger in such models. Consequently the probability that a quasar is lensed by intervening galaxies increases appreciably in a Lambda dominated universe, and can be used as a test to constrain the value of OmegaLambda [72, 71, 192]. Following [73, 26, 37] we give below the probability of a quasar at redshift zs being lensed relative to the fiducial Einstein-de Sitter model (Omegam = 1)

Equation 32 (32)

where d (z1, z2) is a generalization of the angular distance dA = dL(1 + z)-2 discussed in Section 4.5:

Equation 33 (33)


Equation 34 (34)

and S(eta12) is defined as follows, S(eta12) = sin(eta12) if kappa = 1 (Omegatotal > 1), S(eta12) = sinh(eta12) if kappa = -1 (Omegatotal < 1), S(eta12) = eta12 if kappa = 0 (Omegatotal = 1). In Fig 5 we show the lensing probability P(lens) for the spatially flat universe Omegam + OmegaLambda = 1. A large increase in the lensing probability over the fiducial Omegam = 1 value is clearly seen in models with low Omegam (high OmegaLambda). (For a broader analysis of parameter space see [26].)

Figure 5

Figure 5. The lensing probability P(lens) evaluated relative to the fiducial case Omegam = 1 is shown as a function of OmegaLambda for flat cosmological models Omegam + OmegaLambda = 1. The source redshift is taken at zs = 1, 2, 3 respectively.

Turning now to the observational situation, at the time of writing the best observational estimates give a 2sigma upper bound OmegaLambda < 0.66 obtained from multiple images of lensed quasars [111, 112, 136]. Since radio sources are not plagued by some of the systematic errors arising in an optical search (notably extinction in the lens galaxy and the quasar discovery process) a search involving radio selected lenses can yield useful complementary information to optical searches [60]. Recent work by Falco et al (1998) gives OmegaLambda < 0.73 which is only marginally consistent with optical estimates, a combined analysis of optical and radio data yields a slightly more conservative upper bound OmegaLambda < 0.62 at the 2sigma level (for flat universes) [60]. (Constraints on OmegaLambda from both lensing and Type 1a Supernovae are discussed in [201]; also see next section. An interesting new method of constraining OmegaLambda from weak lensing in clusters is discussed in [67], also see [12] and section 4.6.) Improved understanding of statistical and systematic uncertainties combined with new surveys and better quality data promise to make gravitational lensing a powerful technique for constraining cosmological parameters and cosmological world models.

5 For instance a galaxy at redshift z = 3 will appear 9 times brighter in a flat matter dominated universe than it will in de Sitter space (see Fig 4). Back.

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